Inferring hidden symmetries of exotic magnets from detecting explicit order parameters

An unconventional magnet may be mapped onto a simple ferromagnet by the existence of a high-symmetry point. Knowledge of conventional ferromagnetic systems may then be carried over to provide insight into more complex orders. Here we demonstrate how an unsupervised and interpretable machine-learning approach can be used to search for potential high-symmetry points in unconventional magnets without any prior knowledge of the system. The method is applied to the classical Heisenberg-Kitaev model on a honeycomb lattice, where our machine learns the transformations that manifest its hidden $O\left(3\right)$ symmetry, without using data of these high-symmetry points. Moreover, we clarify that, in contrast to the stripy and zigzag orders, a set of $D_2$ and $D_{2h}$ ordering matrices provides a more complete description of the magnetization in the Heisenberg-Kitaev model. In addition, our machine also learns the local constraints at the phase boundaries, which manifest a subdimensional symmetry. This paper highlights the importance of explicit order parameters to many-body spin systems and the property of interpretability for the physical application of machine-learning techniques.

Figure 1

Depiction of the honeycomb lattice and the $D_{2h}$ and $D_2$ magnetic cell, which contain eight spins and two sectors marked by $A$ (blue) and $B$ (red). This choice of magnetic cell fits zigzag and stripy patterns along different directions and also applies to states at the hidden $O\left(3\right)$ points which cannot be captured by a four-site zigzag or stripy cell. $x,y,\text{and}z$ label the three distinct bonds in the Kitaev interaction.

Figure 2

The Fiedler vector acts as the phase diagram of the Heisenberg-Kitaev model. (a) Gradients in Fiedler vector entries reflect the clustering of the graph. The plateaus indicate stable phases, and the jumps signal phase transitions. The phases are interpreted in Sec. 4 and are labeled following the common convention: AFM, antiferromagnet; ZZ, zigzag; FM, ferromagnet; ST, stripy. In addition, the ST and ZZ region are also marked according to the $D_2$ and $D_{2h}$ magnetization measured in Fig. 3. The inner panel shows a circular representation of the phase diagram. (b) Another partitioning with removing data near the high-symmetry points $\phi =arctan(-2)\sim 0.65\pi \text{and}1.65\pi$ (indicated by the dashed lines; the graph is not shown), to demonstrate that data of these special points are not needed for revealing the hidden $O\left(3\right)$ symmetry. The partitioning is reflected by contrasts between Fiedler vector entries, rather than the absolute values. Panels (a) and (b) lead to the same topology of the phase diagram.

Figure 3

Measurements of order parameters. The FM, AFM, $D_2$, and $D_{2h}$ magnetization are measured as a function of $\phi$ at low temperature $T={10}^{-3}\sqrt{J^2+K^2}$. In each phase, the respective magnetization saturates to unity ($M=〈|\frac{1}{N_{\mathrm{cell}}}{\sum }_{\mathrm{cell}}\vec{M}|〉=1$), while others vanish, where $\vec{M}$ is the ordering moment in one magnetic cell, ${\sum }_{\mathrm{cell}}$ sums over magnetic cells, and $\langle \cdots \rangle$ denotes the ensemble average. The small residual moments at $\phi =0.75\pi \text{and}1.75\pi$ are finite-size effects. At these points, the classical ground states form decoupled FM and AFM Ising chains with a subextensive degeneracy (Sec. 6).

Figure 4

Static spin-structure factor, $S\left(\vec{K}\right)$, for the ST and $D_2$ (left) and ZZ and $D_{2h}$ order (right). The orange and gray hexagon denote the second and first honeycomb Brillouin zone, respectively, and high-symmetry points are indicated. $S\left(\vec{K}\right)=〈\frac{1}{N}{\sum }_{ij}{\vec{S}}_i·{\vec{S}}_j{e}^{i\vec{K}·({\vec{r}}_i-{\vec{r}}_j)}〉$, where ${\vec{r}}_i$ is the position of a spin at site $i$, and a nearest-neighbor bond of the honeycomb lattice is set to unit length.

Figure 5

Configurations of an arbitrary $D_2$ and $D_{2h}$ state. The spin ${\vec{S}}_{A_1}={\left(S_x{S}_y{S}_z\right)}^{\mathrm{T}}$ is used as the reference spin, while orientations of other spins are determined according to the respective ordering matrices. Compared to stripy and zigzag orders, which are staggered arrangements of $\pm \vec{S}$, the sign flip in a $D_2$ and $D_{2h}$ pattern can occur at individual components. In special cases ${\vec{S}}_{A_1}={(00\pm 1)}^{\mathrm{T}}$, these patterns are equivalent to the $Z$-type zigzag and stripy patterns in Fig. 6, with a reduced four-site magnetic cell $\{A_1,A_2,A_3,B_3\}$. When choosing ${\vec{S}}_{A_1}={(\pm 100)}^{\mathrm{T}}$ and ${(0\pm 10)}^{\mathrm{T}}$, $X$- and $Y$-type zigzag and stripy states will be realized, where the magnetic cells are given by $\{A_1,A_2,A_3,B_1\}$ and $\{A_2,A_3,B_1,B_2\}$, respectively. In general cases, the $D_{2h}$ ($D_2$) and zigzag (stripy) orders are different, and the magnetic cell cannot be reduced to four sites.

Figure 6

Representative configurations of a stripy (ST) and zigzag (ZZ) order. White ($\vec{S}$) and black ($-\vec{S}$) cycles denote opposite spins. The corresponding magnetization can be defined as $M_{\mathrm{ST}}=〈\left|\frac{1}{N_{\mathrm{cell}}}{\sum }_{\mathrm{cell}}({\vec{S}}_1+{\vec{S}}_2-{\vec{S}}_3-{\vec{S}}_4)\right|〉$ and $M_{\mathrm{ZZ}}=〈\left|\frac{1}{N_{\mathrm{cell}}}{\sum }_{\mathrm{cell}}({\vec{S}}_1-{\vec{S}}_2+{\vec{S}}_3-{\vec{S}}_4)\right|〉$, respectively, where the numbers label the four sublattices. In general, $\vec{S}$ may point to arbitrary directions. However, in the ground states of the Heisenberg-Kitaev model, the realization of these above configurations will be accompanied by $\vec{S}={(00\pm 1)}^{\mathrm{T}}$. We hence refer to them as $Z$ type. Such states are present in the intersection of zigzag (stripy) and $D_{2h}$ ($D_2$) manifolds.

Figure 7

Distribution of spin orientations for states in the ZZ or $D_{2h}$ and ST or $D_2$ phases away (left) and at (right) the hidden $O\left(3\right)$ points, at a low temperature $T=0.001$.

Figure 8

Magnetization as a function of temperature at the $O\left(3\right)$ points, with $\phi \approx 0.65\pi$ for the zigzag (ZZ) and $D_{2h}$ orders and $\phi \approx 1.65\pi$ for the stripy (ST) and $D_2$ orders. The $D_{2h}$ (ZZ) and $D_2$ (ST) curves show the same behavior, as the Heisenberg-Kitaev model is symmetric under a sublattice transformation $J\to -J,K\to -K$, and meanwhile $S_i\to -S_i$ for either of the honeycomb sublattices.

Figure 9

The $C_{\mu \nu }$ matrix learned by a rank-2 TK-SVM with a four-spin triad cluster (inner panel) at the boundary point $\phi =\frac{3\pi }{4}$. The axes iterate over spin indices $(i,j)$ and spin components $(\alpha ,\beta )$ in a lexicographical order, from bottom (left) to top (right). The spin indices divide the $C_{\mu \nu }$ matrix into $9\times 9$ sub-blocks. Nonvanishing entries in a block represent the form of correlations between quadratic components $S_i^{\alpha }{S}_j^{\beta }$ and $S_{i^{\prime }}^{{\alpha }^{\prime }}{S}_{j^{\prime }}^{{\beta }^{\prime }}$. Blocks with $i=j$ and $i^{\prime }=j^{\prime }$ lead to constants owing to the trivial normalization $|\vec{S}|=1$. Other blocks correspond to the local constraints $G_1$ and $G_2$. The pattern learned for $\phi =\frac{7\pi }{4}$ (not shown) has a similar structure with sign flips in certain entries.

Figure 10

Local constraints at $\phi =\frac{3\pi }{4},\frac{7\pi }{4}$ as a function of temperature. $G_1$ and $G_2$ satisfy Eqs. (9) and (10) in the ground state. (The $G_2$ curves at the two $\phi$ values overlap.)

Figure 11

Representative classical ground-state configuration at $\phi =\frac{3\pi }{4},\frac{7\pi }{4}$. The system forms ferromagnetic (a) or antiferromagnetic (b) Ising chains. The subdimensional symmetry leads to a classical subextensive degeneracy by flipping one entire chain of spins.

Figure 12

The $C_{\mu \nu }$ matrices learned by a rank-1 TK-SVM in the ST or $D_2$ and ZZ or $D_{2h}$ phases. Each entry represents a correlation between two spin components defined by the weighted sum of the support vectors. Results of an eight-spin cluster ($2\times 2$ honeycomb unit cells), which is the minimal unit of the $D_2$ and $D_{2h}$ order, are shown for demonstration. From bottom to top, the vertical axis is labeled in the same convention as in the lattice Fig. 1. The same labeling applies to the horizontal axis from left to right. The interpretation of these patterns leads to the respective ordering matrices in Table 1.

Figure 13

Fiedler vectors obtained with different choices of ${\rho }_c$. In all cases, where ${\rho }_c$ is large enough to set a characteristic scale “$\gg 1$” for the reduced $\rho$ criterion Eq. (B1), the clustering is evident and robust. The profound jumps at $\phi =\frac{\pi }{2},\frac{3\pi }{4},\frac{3\pi }{2},\frac{7\pi }{4}$ correspond to phase boundaries, as they do not belong to any plateaus (stable phases). A case of small ${\rho }_c=0.1$ is also included for comparison. ${\rho }_c=100$ is used in the main text.

Figure 14

Representative blocks of the $C_{\mu \nu }$ matrices of the ZZ or $D_{2h}$ phase learned by a rank-2 TK-SVM with the eight-spin $D_{2h}$ magnetic cell, away from (a) and at (b) the $O\left(3\right)$ point. Blocks are labeled by the spin indices $(i,j)$. Nonvanishing entries in a block correspond to correlations between quadratic components $S_i^{\alpha }{S}_j^{\beta }$ and $S_{i^{\prime }}^{{\alpha }^{\prime }}{S}_{j^{\prime }}^{{\beta }^{\prime }}$. Negative elements in the (0,0) block reflect the spin normalization $|\vec{S}|=1$. Nontrivial entries in (a) are the diagonal ones in each $9\times 9$ sub-block.